reserve Al for QC-alphabet,
     PHI for Consistent Subset of CQC-WFF(Al),
     PSI for Subset of CQC-WFF(Al),
     p,q,r,s for Element of CQC-WFF(Al),
     A for non empty set,
     J for interpretation of Al,A,
     v for Element of Valuations_in(Al,A),
     m,n,i,j,k for Element of NAT,
     l for CQC-variable_list of k,Al,
     P for QC-pred_symbol of k,Al,
     x,y for bound_QC-variable of Al,
     z for QC-symbol of Al,
     Al2 for Al-expanding QC-alphabet;
reserve J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th8:
 for p,q,A,J,v holds (J,v |= p or J,v |= q) iff J,v |= p 'or' q
proof
  let p,q,A,J,v;
  thus (J,v |= p or J,v |= q) implies J,v |= p 'or' q
  proof
    assume J,v |= p or J,v |= q;
    then not J,v |= 'not' p or not J,v |= 'not' q by VALUAT_1:17;
    then not J,v |= 'not' p '&' 'not' q by VALUAT_1:18;
    then J,v |= 'not' ('not' p '&' 'not' q) by VALUAT_1:17;
    hence thesis by QC_LANG2:def 3;
 end;
 thus J,v |= p 'or' q implies (J,v |= p or J,v |= q)
 proof
   assume J,v |= p 'or' q;
   then J,v |= 'not' ('not' p '&' 'not' q) by QC_LANG2:def 3;
   then not J,v |= 'not' p or not J,v |= 'not' q by VALUAT_1:17,18;
   hence J,v |= p or J,v |= q by VALUAT_1:17;
 end;
end;
